1,2

This sequence is an involutory (period 2) permutation of the natural numbers. Its graph is symmetric in the line a(n) = n. It fixes all numbers with self-conjugate prime signatures (A384084), including 1 and the primes (A000040). It exchanges prime squares (A001248) and products of two distinct primes (A006881). It exchanges powers of 2 (A000079) with primorials (A002110).

The implied partition corresponding to k is the partition of bigomega(k) (A001222) formed by the prime exponents. For example, bigomega(18) = 3, which is partitioned as 2 + 1, because 18 = (3^2)*(2^1). Because this is a self-conjugate partition, a(18) = 18. Similarly, bigomega(42) = 3, which is partitioned as 1 + 1 + 1, because 42 = (2^1)*(3^1)*(7^1). The conjugate partition is 3, so a(42) is the cube of a prime.

sig[n_] := Sort[FactorInteger[n][[;; , 2]], Greater]; conjp[p_] := ResourceFunction["ConjugatePartition"][p]; a[1] = 1; a[n_] := a[n] = Module[{p = conjp[sig[n]], k = 2}, While[!FreeQ[Array[a, n-1], k] || sig[k] != p, k++]; k]; Array[a, 100] (* Amiram Eldar, Jun 04 2025 *)

Cf. A000040, A000079, A001248, A002110, A006881, A384084.

Sequence in context: A371343 A166166 A106452 * A254118 A056895 A383723

Adjacent sequences: A384561 A384562 A384563 * A384565 A384566 A384567

Hal M. Switkay, Jun 03 2025

approved